Jarzynski equality and the second law of thermodynamics beyond the weak-coupling limit: The quantum Brownian oscillator

We consider a time-dependent quantum linear oscillator coupled to a bath at an arbitrary strength. We then introduce a generalized Jarzynski equality (GJE) which includes the terms reflecting the system-bath coupling. This enables us to study systematically the coupling effect on the linear oscillator in a non-equilibrium process. This is also associated with the second law of thermodynamics beyond the weak-coupling limit. We next take into consideration the GJE in the classical limit. By this generalization we show that the Jarzynski equality in its original form can be associated with the second law, in both quantal and classical domains, only in the vanishingly small coupling regime.Comment: 7 figure

An analytic study of the ionization from an ultrathin quantum well in a weak electrostatic field

We consider the time evolution of a particle bound by an attractive one-dimensional delta-function potential (at x = 0) when a uniform electrostatic field (F) is applied. We explore explicit expressions for the time-dependent wavefunction \psi_F(x,t) and the ionization probability {\mathcal{P}}(t), respectively, in the weak-field limit. In doing so, \psi_F(0,t) is a key element to their evaluation. We obtain a closed expression for \psi_F(0,t) which is an excellent approximation of the exact result being a numerical solution of the Lippmann-Schwinger integral equation. The resulting probability density |\psi_F(0,t)|^2, as a simple alternative to {\mathcal{P}}(t), is also in good agreement to its counterpart from the exact one. In doing this, we also find a new and useful integral identity of the Airy function.Comment: 3 figure

Electrostatic-field-induced dynamics in an ultrathin quantum well

We consider the time evolution of a particle subjected to both a uniform electrostatic field F and a one-dimensional delta-function potential well. We derive the propagator K_F(x,t|x',0) of this system, directly leading to the wavefunction psi_F(x,t), in which its essential ingredient K_F(0,t|0,0), accounting for the ionization-recombination in the bound-continuum transition, is exactly expressed in terms of the multiple hypergeometric functions F(z_1,z_2,...,z_n). And then we obtain the ingredient K_F(0,t|0,0) in an appropriate approximation scheme, expressed in terms of the generalized hypergeometric functions p_F_q(z) being much more transparent to physically interpret and much more accessible in their numerical evaluation than the functions F(z_1,z_2,...,z_n).Comment: 24 pages, 1 figur

Moving quantum agents in a finite environment

We investigate an all-quantum-mechanical spin network, in which a subset of spins, the $K$ ``moving agents'', are subject to local and pair unitary transformations controlled by their position with respect to a fixed ring of $M$ ``environmental''-spins. We demonstrate that a ``flow of coherence'' results between the various subsystems. Despite entanglement between the agents and between agent and environment, local (non-linear) invariants may persist, which then show up as fascinating patterns in each agent's Bloch-sphere. Such patterns disappear, though, if the agents are controlled by different rules. Geometric aspects thus help to understand the interplay between entanglement and decoherence.Comment: REVTEX, to appear in Proceedings of Decoherence Workshop, Bielefeld, 199

Non-negative Wigner-like distributions and Renyi-Wigner entropies of arbitrary non-Gaussian quantum states: The thermal state of the one-dimensional box problem

In this work, we consider the phase-space picture of quantum mechanics. We then introduce non-negative Wigner-like (operational) distributions \widetilde{\mathcal W}_{rho;alpha}(x,p) corresponding to the density operator \hat{rho} and being proportional to {W_{rho^(alpha/2)}(x,p)}^2, where W_{rho}(x,p) denotes the usual Wigner function. In doing so, we utilize the formal symmetry between the purity measure Tr(rho^2) and its Wigner representation (2 pi hbar) \int dx dp {W_{rho}(x,p)}^2 and then consider, as a generalization, such symmetry between the fractional moment Tr(\hat{rho}^{alpha}) and its Wigner representation (2 pi hbar) \int dx dp {W_{rho^{alpha/2}}(x,p)}^2. Next, we create a framework that enables explicit evaluation of the Renyi-Wigner entropies for the classical-like distributions \widetilde{\mathcal W}_{rho;alpha}(x,p). Consequently, a better understanding of some non-Gaussian features of a given state rho will be given, by comparison with the Gaussian state rho_G defined in terms of its Wigner function W_{rho_G}(x,p) and essentially determined by its purity measure T(rho_G)^2 alone. To illustrate the validity of our framework, we evaluate the distributions \widetilde{\mathcal W}_{beta;alpha}(x,p) corresponding to the (non-Gaussian) thermal state rho_{\beta} of a single particle confined by a one-dimensional infinite potential well with either the Dirichlet or Neumann boundary condition and then analyze the resulting Renyi entropies. Our phase-space approach will also contribute to a deeper understanding of non-Gaussian states and their properties either in the semiclassical limit (hbar \to 0) or in the high-temperature limit (beta \to 0), as well as enabling us to systematically discuss the quantal-classical Second Law of Thermodynamics on the single footing.Comment: 27 pages, 6 figure

Quantum chaos in small quantum networks

We study a 2-spin quantum Turing architecture, in which discrete local rotations \alpha_m of the Turing head spin alternate with quantum controlled NOT-operations. We show that a single chaotic parameter input \alpha_m leads to a chaotic dynamics in the entire Hilbert space. The instability of periodic orbits on the Turing head and `chaos swapping' onto the Turing tape are demonstrated explicitly as well as exponential parameter sensitivity of the Bures metric.Comment: Accepted for publication in JMO (quantum information special issue, Vol 47), 3 figure

The Clausius inequality beyond the weak coupling limit: The quantum Brownian oscillator revisited

We consider a quantum linear oscillator coupled at an arbitrary strength to a bath at an arbitrary temperature. We find an exact closed expression for the oscillator density operator. This state is non-canonical but can be shown to be equivalent to that of an uncoupled linear oscillator at an effective temperature T_{eff} with an effective mass and an effective spring constant. We derive an effective Clausius inequality delta Q_{eff} =< T_{eff} dS, where delta Q_{eff} is the heat exchanged between the effective (weakly coupled) oscillator and the bath, and S represents a thermal entropy of the effective oscillator, being identical to the von-Neumann entropy of the coupled oscillator. Using this inequality (for a cyclic process in terms of a variation of the coupling strength) we confirm the validity of the second law. For a fixed coupling strength this inequality can also be tested for a process in terms of a variation of either the oscillator mass or its spring constant. Then it is never violated. The properly defined Clausius inequality is thus more robust than assumed previously.Comment: 30 pages, 7 figure

Comment on the quantum nature of angular momentum using a coupled-boson representation

A simple approach for understanding the quantum nature of angular momentum and its reduction to the classical limit is presented based on Schwinger's coupled-boson representation. This approach leads to a straightforward explanation of why the square of the angular momentum in quantum mechanics is given by j(j+1) instead of just j^2, where j is the angular momentum quantum number

Ab initio relaxation times and time-dependent Hamiltonians within the steepest-entropy-ascent quantum thermodynamic framework

Quantum systems driven by time-dependent Hamiltonians are considered here within the framework of steepest-entropy-ascent quantum thermodynamics (SEAQT) and used to study the thermodynamic characteristics of such systems. In doing so, a generalization of the SEAQT framework valid for all such systems is provided, leading to the development of an ab initio physically relevant expression for the intra-relaxation time, an important element of this framework and one that had as of yet not been uniquely determined as an integral part of the theory. The resulting expression for the relaxation time is valid as well for time-independent Hamiltonians as a special case and makes the description provided by the SEAQT framework more robust at the fundamental level. In addition, the SEAQT framework is used to help resolve a fundamental issue of thermodynamics in the quantum domain, namely, that concerning the unique definition of process-dependent work and heat functions. The developments presented lead to the conclusion that this framework is not just an alternative approach to thermodynamics in the quantum domain but instead one that uniquely sheds new light on various fundamental but as of yet not completely resolved questions of thermodynamics.Comment: Accepted for publication in Physical Review

Renyi-alpha entropies of quantum states in closed form: Gaussian states and a class of non-Gaussian states

In this work, we study the Renyi-alpha entropies S_{alpha}(\hat{rho}) = (1 - alpha)^{-1} \ln{Tr(\hat{rho}^{alpha})} of quantum states for N bosons in the phase-space representation. With the help of the Bopp rule, we derive the entropies of Gaussian states in closed form for positive integers alpha = 2,3,4, ... and then, with the help of the analytic continuation, acquire the closed form also for real values of alpha > 0. The quantity S_2(\hat{rho}), primarily studied in the literature, will then be a special case of our finding. Subsequently we acquire the Renyi-alpha entropies, with positive integers alpha, in closed form also for a specific class of the non-Gaussian states (mixed states) for N bosons, which may be regarded as a generalization of the eigenstates |n> (pure states) of a single harmonic oscillator with n >= 1, in which the Wigner functions have negative values indeed. Due to the fact that the dynamics of a system consisting of $N$ oscillators is Gaussian, our result will contribute to a systematic study of the Renyi-alpha entropy dynamics when the current form of a non-Gaussian state is initially prepared.Comment: Accepted for publication in Physical Review